Chapter 2: Diffraction

2.13. Lattice Determination with HOLZ#

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part of

MSE672: Introduction to Transmission Electron Microscopy

by Gerd Duscher, Spring 2023

Microscopy Facilities
Joint Institute of Advanced Materials
Materials Science & Engineering
The University of Tennessee, Knoxville

Background and methods to analysis and quantification of data acquired with transmission electron microscopes

2.13.1. Load relevant python packages#

2.13.1.1. Check Installed Packages#

import sys
from pkg_resources import get_distribution, DistributionNotFound

def test_package(package_name):
    """Test if package exists and returns version or -1"""
    try:
        version = get_distribution(package_name).version
    except (DistributionNotFound, ImportError) as err:
        version = '-1'
    return version

# Colab setup ------------------
if 'google.colab' in sys.modules:
    !pip install pyTEMlib -q
# pyTEMlib setup ------------------
else:
    if test_package('pyTEMlib') < '0.2022.2.5':
        print('installing pyTEMlib')
        !{sys.executable} -m pip install  --upgrade pyTEMlib -q
# ------------------------------
print('done')

done

2.13.2. Import numerical and plotting python packages#

Import the python packages that we will use:

Beside the basic numerical (numpy) and plotting (pylab of matplotlib) libraries,

and some libraries from the book

  • kinematic scattering library.

  • file_tools library

# import matplotlib and numpy
#                       use "inline" instead of "notebook" for non-interactive plots
%matplotlib  notebook
import matplotlib.pyplot as plt
import numpy as np

%gui 
# additional package 
import  itertools 
import scipy.constants as const

import ipywidgets as ipyw

# Import libraries from pyTEMlib
import pyTEMlib
import pyTEMlib.kinematic_scattering as ks         # Kinematic sCattering Library
                             # Atomic form factors from Kirklands book

### And we use the image tool library of Quantifit
import pyTEMlib.file_tools as ft
import pyTEMlib

__notebook_version__ = '2022.02.21'

print('pyTEM version: ', pyTEMlib.__version__)
print('notebook version: ', __notebook_version__)

Symmetry functions of spglib enabled
Using kinematic_scattering library version {_version_ }  by G.Duscher
pyTEM version:  0.2023.2.0
notebook version:  2022.02.21

2.13.3. Define crystal#

### Please choose another crystal like: Silicon, Aluminium, GaAs , ZnO
atoms = ks.structure_by_name('silicon')

2.14. Plot the unit cell#

## Just to be sure the crystal structure is right
from ase.visualize.plot import plot_atoms

plot_atoms(atoms, radii=0.3, rotation=('0x,4y,0z'))

<AxesSubplot:>

2.14.1. Load Diffraction Pattern#

image = plt.imread("images/Zuo-HOLZ-experiment.jpg")
plt.figure()
plt.imshow(image);

2.15. Parameters for Diffraction Calculation#

Please note that we are using a rather small number of reflections: the maximum number of hkl is 1

tags = {}
tags['acceleration_voltage_V'] = 98.8*1000.0 #V

tags['zone_hkl'] = np.array([-2,2,1])  # incident neares zone axis: defines Laue Zones!!!!
tags['mistilt']  = np.array([-0.0,-0.0,-0.0])  # mistilt in degrees

tags['Sg_max'] = .2 # 1/nm  maximum allowed excitation error ; This parameter is related to the thickness
tags['hkl_max'] = 9   # Highest evaluated Miller indices
tags['convergence_angle_nm'] = 1.9
atoms.info['experimental'] = tags

2.15.1. Function for Coverting Hough Coordinates to Line Coordinates#

def Hough2Lines(tags):
    angle = np.radians(tags['plot_rotation']) #mrad
    # Calculate nearest point of HOLZ and Kikuchi lines
    
    L = tags['convergence_angle_A-1']*2
    if L < 0.1:
        L=3
    theta = tags['HOLZ']['theta']-angle
    gd_length = tags['HOLZ']['distance'] 
    
    g_closest = np.ones((len(theta),2))
    g_closest[:,1] = np.cos(theta)
    g_closest[:,0] = np.sin(theta)
    
    g_closest =  g_closest*(gd_length/np.linalg.norm(g_closest, axis =1))[:,np.newaxis]
    
    g_line = np.zeros(g_closest.shape)
    g_line[:,0] = np.cos(np.pi-theta)
    g_line[:,1] = np.sin(np.pi-theta)
    

    line_start = g_closest+g_line*L
    line_end = g_closest-g_line*L
    lines = np.zeros((g_line.shape[0],5))
    lines[:,0] = line_start[:,0]
    lines[:,1] = line_end[:,0]
    lines[:,2] = line_start[:,1]
    lines[:,3] = line_end[:,1]
    lines[:,4] = tags['HOLZ']['intensities']#tags['allowed']['intensities']
    
    lines[tags['allowed']['ZOLZ'],4] = tags['allowed']['intensities'][tags['allowed']['ZOLZ']]/tags['allowed']['intensities'].max()*4
    return lines, g_closest

2.15.2. Initial Overlay#

atoms.info['experimental']['acceleration_voltage_V'] = 99.2*1000.0 #V

tags['zone_hkl'] = np.array([-2,2,1])  # incident neares zone axis: defines Laue Zones!!!!
tags['mistilt']  = np.array([-0.0,-0.0,-0.0])  # mistilt in degrees

tags['Sg_max'] = .04 # 1/nm  maximum allowed excitation error ; This parameter is related to the thickness
tags['hkl_max'] = 9   # Highest evaluated Miller indices
tags['convergence_angle_A-1'] = 1.9
atoms.info['experimental'] = tags

ks.kinematic_scattering(atoms, False)

tagsD = atoms.info['diffraction']
tagsD['convergence_angle_A-1'] = .19
tagsD['plot_rotation'] = 162
tagsD['plot image FOV'] = .516
tagsD['plot shift x'] = 0.0065
tagsD['plot shift y'] = .043

lines, g_closest = Hough2Lines(tagsD)

# we sort them by order of Laue zone
ZOLZ = tagsD['allowed']['ZOLZ']
HOLZ = tagsD['allowed']['HOLZ']

l = -tagsD['plot image FOV']/2+tagsD['plot shift x']
r = tagsD['plot image FOV']/2+tagsD['plot shift x']
t = -tagsD['plot image FOV']/2+tagsD['plot shift y']
b = tagsD['plot image FOV']/2+tagsD['plot shift y']

plt.figure()#
plt.imshow(image, extent=(r,l,t,b))
plt.scatter(g_closest[:,0],g_closest[:,1])
#plt.scatter(line_start[:,0],line_start[:,1],c='g')
#plt.scatter(line_end[:,0],line_end[:,1],c='g')
plt.plot((lines[HOLZ,0],lines[HOLZ,1]),(lines[HOLZ,2],lines[HOLZ,3]),c='b')
plt.plot((lines[ZOLZ,0],lines[ZOLZ,1]),(lines[ZOLZ,2],lines[ZOLZ,3]),c='c')
zero_disk = plt.Circle((tagsD['plot shift x']*2.1, tagsD['plot shift y']*0.7), tagsD['convergence_angle_A-1'],
                       color='r', fill=False)
plt.scatter([0],[0],c='red')
plt.gca().add_artist(zero_disk)
s = 0.2
plt.gca().set_xlim(-s,s)
plt.gca().set_ylim(-s,s)
plt.show()

2.15.3. Plotting with mistilt included and objective stigmation compensated#

# -----Input----
O_stig = 1.07
# --------------

atoms.info['experimental']['acceleration_voltage_V'] = 99.2*1000.0 #V

tags['zone_hkl'] = np.array([-2,2,1])  # incident neares zone axis: defines Laue Zones!!!!
tags['mistilt']  = np.array([-0.05,-0.05,-0.03])  # mistilt in degrees

tags['Sg_max'] = .02 # 1/nm  maximum allowed excitation error ; This parameter is related to the thickness
tags['hkl_max'] = 9   # Highest evaluated Miller indices
tags['unit_cell'][0,0]=tags['unit_cell'][1,1]*1.00
tags['unit_cell'][2,2]=tags['unit_cell'][1,1]*1.00
tags['unit_cell'][2,1] = .00
tags['unit_cell'] = tags['unit_cell']*1.00

tags['convergence_angle_A-1'] = 1.9
atoms.info['experimental'] = tags

ks.kinematic_scattering(atoms, False)

tagsD = atoms.info['diffraction']
tagsD['convergence_angle_A-1'] = .19
tagsD['plot_rotation'] = 162
tagsD['plot image FOV'] = .516
tagsD['plot shift x'] = 0.006
tagsD['plot shift y'] = .043

lines, g_closest = Hough2Lines(tagsD)

# we sort them by order of Laue zone
ZOLZ = tagsD['allowed']['ZOLZ']
HOLZ = tagsD['allowed']['HOLZ']

l = -tagsD['plot image FOV']/2*O_stig + tagsD['plot shift x']
r = tagsD['plot image FOV']/2*O_stig + tagsD['plot shift x']
t = -tagsD['plot image FOV']/2+tagsD['plot shift y']
b = tagsD['plot image FOV']/2+tagsD['plot shift y']

plt.figure()#
plt.imshow(image, extent=(r,l,t,b))
plt.scatter(g_closest[:,0],g_closest[:,1])
#plt.scatter(line_start[:,0],line_start[:,1],c='g')
#plt.scatter(line_end[:,0],line_end[:,1],c='g')
plt.plot((lines[HOLZ,0],lines[HOLZ,1]),(lines[HOLZ,2],lines[HOLZ,3]),c='b')
plt.plot((lines[ZOLZ,0],lines[ZOLZ,1]),(lines[ZOLZ,2],lines[ZOLZ,3]),c='c')
zero_disk = plt.Circle((tagsD['plot shift x']*2.1, tagsD['plot shift y']*0.7), tagsD['convergence_angle_A-1'],
                       color='r', fill=False)
plt.scatter([0],[0],c='red')
plt.gca().add_artist(zero_disk)
s = 0.2
plt.gca().set_xlim(-s,s)
plt.gca().set_ylim(-s,s);

2.15.4. Plotting more HOLZ lines with intensity#

In the image above, all major lines are reproduced.

On the bottom of the HOLZ plot, however, the faint lines are not there.

Increase the maximum hkl in the simulation to see how well we did

# -----Input----
Sg_max = .02 # 1/nm  maximum allowed excitation error ; This parameter is related to the thickness
maximum_hkl = 16   # Highest evaluated Miller indices
# --------------

O_stig = 1.07

atoms.info['experimental']['acceleration_voltage_V'] = 99.2*1000.0 #V

tags['zone_hkl'] = np.array([-2,2,1])  # incident neares zone axis: defines Laue Zones!!!!
tags['mistilt']  = np.array([0,0,0]) #-0.05,-0.07,-0.03])  # mistilt in degrees

tags['Sg_max'] = Sg_max # 1/nm  maximum allowed excitation error ; This parameter is related to the thickness
tags['hkl_max'] = maximum_hkl   # Highest evaluated Miller indices

tags['convergence_angle_A-1'] = 1.9
atoms.info['experimental'] = tags

ks.kinematic_scattering(atoms, False)

tagsD = atoms.info['diffraction']
tagsD['convergence_angle_A-1'] = .19
tagsD['plot_rotation'] = 162
tagsD['plot image FOV'] = .516
tagsD['plot shift x'] = 0.006
tagsD['plot shift y'] = .043

lines, g_closest = Hough2Lines(tagsD)

# we sort them by order of Laue zone
ZOLZ = tagsD['allowed']['ZOLZ']
HOLZ = tagsD['allowed']['HOLZ']

l = -tagsD['plot image FOV']/2*O_stig + tagsD['plot shift x']
r = tagsD['plot image FOV']/2*O_stig + tagsD['plot shift x']
t = -tagsD['plot image FOV']/2+tagsD['plot shift y']
b = tagsD['plot image FOV']/2+tagsD['plot shift y']

plt.figure()
plt.imshow(image, extent=(r,l,t,b))
plt.plot((lines[HOLZ,0],lines[HOLZ,1]),(lines[HOLZ,2],lines[HOLZ,3]),c='b')
plt.plot((lines[ZOLZ,0],lines[ZOLZ,1]),(lines[ZOLZ,2],lines[ZOLZ,3]),c='c')
zero_disk = plt.Circle((tagsD['plot shift x']*2.1, tagsD['plot shift y']*0.7), tagsD['convergence_angle_A-1'],
                       color='r', fill=False)
plt.scatter([0],[0],c='red')
plt.gca().add_artist(zero_disk)
s = 0.2
plt.gca().set_xlim(-s,s)
plt.gca().set_ylim(-s,s);

2.15.5. Save unit cell data#

unit_cell = tags['unit_cell'].copy()

2.15.6. Influence of unit cell deformation#

Introduce a bit of distortion in the unit cell and see what happens

# -----Input----
tags['unit_cell'][0,0]=unit_cell[0,0]+0.00
tags['unit_cell'][2,2]=unit_cell[1,1]+0.00
tags['unit_cell'][2,1] = .00
tags['unit_cell'] = unit_cell*1.0
# --------------


Sg_max = .02 # 1/nm  maximum allowed excitation error ; This parameter is related to the thickness
maximum_hkl = 9   # Highest evaluated Miller indices

O_stig = 1.07

atoms.info['experimental']['acceleration_voltage_V'] = 99.2*1000.0 #V

tags['zone_hkl'] = np.array([-2,2,1])  # incident neares zone axis: defines Laue Zones!!!!
tags['mistilt']  = np.array([0,0,0]) #-0.05,-0.07,-0.03])  # mistilt in degrees

tags['Sg_max'] = Sg_max # 1/nm  maximum allowed excitation error ; This parameter is related to the thickness
tags['hkl_max'] = maximum_hkl   # Highest evaluated Miller indices

tags['convergence_angle_A-1'] = 1.9
atoms.info['experimental'] = tags

ks.kinematic_scattering(atoms, False)

tagsD = atoms.info['diffraction']
tagsD['convergence_angle_A-1'] = .19
tagsD['plot_rotation'] = 162
tagsD['plot image FOV'] = .516
tagsD['plot shift x'] = 0.006
tagsD['plot shift y'] = .043

lines, g_closest = Hough2Lines(tagsD)

# we sort them by order of Laue zone
ZOLZ = tagsD['allowed']['ZOLZ']
HOLZ = tagsD['allowed']['HOLZ']

l = -tagsD['plot image FOV']/2*O_stig + tagsD['plot shift x']
r = tagsD['plot image FOV']/2*O_stig + tagsD['plot shift x']
t = -tagsD['plot image FOV']/2+tagsD['plot shift y']
b = tagsD['plot image FOV']/2+tagsD['plot shift y']

plt.figure()
plt.imshow(image, extent=(r,l,t,b))
plt.plot((lines[HOLZ,0],lines[HOLZ,1]),(lines[HOLZ,2],lines[HOLZ,3]),c='b')
plt.plot((lines[ZOLZ,0],lines[ZOLZ,1]),(lines[ZOLZ,2],lines[ZOLZ,3]),c='c')
zero_disk = plt.Circle((tagsD['plot shift x']*2.1, tagsD['plot shift y']*0.7), tagsD['convergence_angle_A-1'],
                       color='r', fill=False)
plt.scatter([0],[0],c='red')
plt.gca().add_artist(zero_disk)
s = 0.2
plt.gca().set_xlim(-s,s)
plt.gca().set_ylim(-s,s);

2.15.7. Conclusion#

Due to the high angles involved in the scattering but measured at low angles, HOLZ lines provide a very sensitive measurement of experimental and materials parameter.